Draft, May 25, 2016 NUMERICAL COMPUTATION OF GALOIS GROUPS JONATHAN D. HAUENSTEIN, JOSE ISRAEL RODRIGUEZ, AND FRANK SOTTILE Abstract. The Galois/monodromy group of a family of geometric problems or equations is a subtle invariant that encodes the structure of the solutions. Computing monodromy permutations using numerical algebraic geometry gives information about the group, but can only determine it when it is the full symmetric group. We give numerical methods to compute the Galois group and study it when it is not the full symmetric group. One algorithm computes generators while the other gives information on its structure as a permutation group. We illustrate these algorithms with examples using a Macaulay2 package we are developing that relies upon Bertini to perform monodromy computations. AMS Subject Classification 2010: 65H10, 65H20, 14Q15. 1. Introduction Galois groups, which are a pillar of number theory and arithmetic geometry, encode the structure of field extensions. For example, the Galois group of the cyclotomic extension of Q given by the polynomial x4 + x3 + x2 + x + 1 is the cyclic group of order four, and not the full symmetric group. A finite extension L=K, where K has transcendence degree n over C, corresponds to a branched cover f : V ! U of complex algebraic varieties of dimension n, with L the function field of V and K the function field of U. The Galois group of the Galois closure of L=K equals the monodromy group of the branched cover [14, 19]. When U is rational, f : V ! U may be realized as a family of polynomial systems rationally parame- terized by points of U. Applications of algebraic geometry and enumerative geometry are sources of such families. For these, internal structure such as numbers of real solutions and symmetry of the original problem are encoded in the Galois/monodromy group. Computing monodromy is a fundamental operation in numerical algebraic geometry. Com- puting monodromy permutations along randomly chosen loops in the base U was used in [24] to show that several Schubert problems had Galois/monodromy group the full symmetric group. Leaving aside the defect of that computation|the continuation (and hence the mon- odromy permutations) was not certified—this method only computes an increasing sequence of subgroups of the Galois group, and thus only determines the Galois group when it is the full symmetric group. In all other cases, this method lacks a stopping criterion. We offer two additional numerical methods to obtain certifiable information about Galois groups and investigate their efficacy. The first method is easiest to describe when U is a rational curve so that K = C(t), the field of rational functions. Then V is an algebraic curve C equipped with a dominant map f : C ! C whose fiber at t 2 C consists of solutions to a polynomial system that depends upon t. This is a degree k cover outside the branch Research of Hauenstein supported in part by NSF grant ACI-1460032, Sloan Research Fellowship, and Army Young Investigator Program (YIP). Research of Rodriguez supported in part by NSF grant DMS-1402545. Research of Sottile supported in part by NSF grant DMS-1501370. 1 2 J. D. HAUENSTEIN, J. I. RODRIGUEZ, AND F. SOTTILE locus B, which is a finite subset of C. The monodromy group of f : C ! C is generated by permutations coming from loops encircling each branch point. Our second method uses numerical irreducible decomposition of the s-fold fiber product to determine orbits of the monodromy group acting on s-tuples of distinct points in a fiber. When s = k−1, this computes the Galois group. The partial information obtained when s < k−1 may be sufficient to determine the Galois group. We illustrate these methods. The irreducible polynomial x4 − 4x2 + t over C(t) defines a curve C in Cx × Ct whose projectionp pC ! Ct is four-to-one for t 62 B = f0; 4g. The fiber above the point t = 3 is {− 3; −1; 1; 3g. Following these points along a loop in Ct based at t = 3 that encircles the branch point t = 0 gives thep 2-cycle (p−1; 1). A loop encircling the branch point t = 4 gives the product of 2-cycles, (− 3; −1)(1; 3). These permutations generate the Galois group, which is isomorphic to the dihedral group D4 and has order 8. x C y 1 t x 3 4 −1 Figure 1. Curve C over Ct and fiber of C ×Ct C over t = 3. The fiber product C ×Ct C consists of triples (x; y; t), where x and y lie in the fiber of C 4 2 4 2 above t. It is defined in Cx × Cy × Ct by the polynomials x − 4x + t and y − 4y + t. Since (x4 − 4x2 + t) − (y4 − 4y2 + t) = (x − y)(x + y)(x2 + y2 − 4) ; it has three components. One is the diagonal defined by x − y and x4 − 4x2 + t. The off- diagonal consists of two irreducible components, which implies that the action of the Galois 4 2 group G is not two-transitive. One component isp definedp by x + y and x − 4x + t. Its fiber over t = 3 consists of the four ordered pairs (± 3; ∓ 3) and (±1; ∓1), which is an orbit of G acting on orderedp p pairs of solutions. This implies that G acts imprimitively as it fixes the partition {− 3; 3g t {−1; 1g. Thus G ⊂ S4 contains no 3-cycle, so G ⊂ D4. The third 2 2 4 2 component is defined by x + y − 4 and x − 4x + t and its projection to Ct has degree eight. Thus G has an orbit of cardinality eight, which implies jGj ≥ 8, from which we can conclude that G is indeed the dihedral group D4. The systematic study of Galois groups of families of geometric problems and equations coming from applications is in its infancy. Nearly every case we know where the Galois group has been determined exhibits a striking dichotomy (e.g., [7, 14, 23, 24, 25, 26, 28, 34, 36]): either the group acts imprimitively, so that it fails to be 2-transitive, or it is at least (k−2)- transitive in that it contains the alternating group (but is expected to be the full symmetric group). The methods we develop here are being used [26] to further investigate Galois groups and we expect they will help to develop Galois groups as a tool to study geometric problems, including those that arise in applications. NUMERICAL COMPUTATION OF GALOIS GROUPS 3 The paper is structured as follows. Section 2 introduces the background material including permutation groups, Galois groups, fundamental groups, fiber products, homotopy continu- ation, and witness sets. In Section 3, we discuss the method of computing monodromy by determining the branch locus, illustrating this on the classical problem of determining the monodromy group of the 27 lines on a cubic surface. In Section 4, we discuss using fiber products to obtain information about the Galois group, illustrating this method with the monodromy action on the lines on a cubic surface. We further illustrate these methods using three examples from applications in Section 5, and we give concluding remarks in Section 6. 2. Galois groups and numerical algebraic geometry We describe some background, including permutation groups, Galois/monodromy groups, and fundamental groups of hypersurface complements from classical algebraic geometry, as well as the topics from numerical algebraic geometry of homotopy continuation, monodromy, witness sets, fiber products, and numerical irreducible decomposition. 2.1. Permutation groups. Let G ⊂ Sk be a subgroup of the symmetric group on k letters. Then G has a faithful action on[ k] := f1; : : : ; kg. For g 2 G and i 2 [k], write g(i) for the image of i under g. We say that G is transitive if for any i; j 2 [k] there is an element g 2 G with g(i) = j. Every group is transitive on some set, e.g., on itself by left multiplication. The group G has an induced action on s-tuples,[ k]s. The action of G is s-transitive if for any two s-tuples (i1; : : : ; is) and (j1; : : : ; js) each having distinct elements, there is a g 2 G with g(ir) = jr for r = 1; : : : ; s. The full symmetric group Sk is k-transitive and its alternating subgroup Ak of even permutations is (k−2)-transitive. There are few other highly transitive groups. This is explained in [9, x 4] and summarized in the following proposition, which follows from the O'Nan-Scott Theorem [29] and the classification of finite simple groups. Proposition 2.1 (Thm. 4.11 [9]). The only 6-transitive groups are the symmetric and alter- nating groups. The only 4-transitive groups are the symmetric and alternating groups, and the Mathieu groups M11, M12, M23, and M24. All 2-transitive permutation groups are known. Tables 7.3 and 7.4 in [9] list the 2-transitive permutation groups. Suppose that G is transitive on [k]. A block is a subset B of [k] such that for every g 2 G either gB = B or gB \ B = ;. The orbits of a block form a G-invariant partition of [k] into blocks. The group G is primitive if its only blocks are [k] or singletons, otherwise it is imprimitive. Any 2-transitive permutation group is primitive, and primitive permutation groups that are not symmetric or alternating are rare|the set of k for which such a nontrivial primitive permutation group exists has density zero in the natural numbers [9, x 4.9]. 2 Each G-orbit O ⊂ [k] determines a graphΓ O with vertex set [k]|its edges are the pairs in O.